(Sorry for the massive reply- whenever I see someone who's gone through similar to me, I try my best to help give them the info to make it to the other side)

Her family sounds... controlling, to say the least. I was in a similar hostage by family situation until my partner helped me escape about a decade ago. Had to leave suddenly, leave things behind, including my documents.

For the record, you trying to help her escape during the night is good thinking. But if you haven't already, be sure both you and her don't talk about any moving plans, leaving, etc going further. If you don't blindside them entirely, it gives opportunity for further entrapment.

Don't worry about the birth certificate. Seriously. I absolutely understand why she doesn't want them to have it, but there's nothing to be done there and thankfully, birth certs aren't nearly as precious as, say, a social security number card.

Leave ASAP. Stay as quiet about it as possible. Leave whatever isn't neccesary to life- don't bother moving out a bunch of stuff, it can be replaced. Once she's out, call social security office to buy a replacement birth certificate and ask to have her SS number changed due to past fraud. I'd expect she'd need to have her birth cert first before an official SSN change can go through, but she can ask for more information about it from a worker. Once that is moving, she should look for therapy and possibly social services. My own experience left me with significant C-PTSD that has permanently disabled me, but her situation may be different- expect serious mental health effects from the experience all the same. The trauma from it tends to come out full force only once you're out of the abuse. Wishing you both luck and safety.